We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function |K ∩F⊥| n−k for random F ∈ Gn,k and K ⊂ R n a centrally symmetric isotropic convex body. This partially answers a question raised by V. Milman and A. Pajor (see [MP], p.88). In the second part we show that every symmetric convex body has random hi...

We show that learning a convex body in R, given random samples from the body, requires 2 √

Sinclair and Jerrum derived a bound on the mixing rate of time-reversible Markov chains in terms of their conductance. We generalize this result, not assuming time-reversibility and using a weaker notion of conductance. We prove an isoperimetric inequality for subsets of a convex body. These results are combined to simplify an algorithm of Dyer, Frieze and Kannan for approximating the volume of...

A convex body R in a normed d-dimensional space M is called reduced if the M-thickness ∆(K) of each convex body K ⊂ R different from R is smaller than ∆(R). We present two characterizations of reduced polytopes in M. One of them is that a convex polytope P ⊂ M is reduced if and only if through every vertex v of P a hyperplane strictly supporting P passes such that the M-width of P in the perpen...

A random polytope, Kn, is the convex hull of n points chosen randomly, independently, and uniformly from a convex body K^R. It is shown here that, with high probability, Kn can be obtained by taking the convex hull of m = o(n) points chosen independently and uniformly from a small neighbourhood of the boundary of K. §

The normal cycle TK associated with a convex body K ⊂ Rn is a current which in principle contains complete information about K. It is known that if a sequence of convex bodies Ki, i ∈ N, converges to a convex body K in the Hausdorff metric, then the associated normal cycles TKi converge to TK in the dual flat seminorm. We give a quantitative improvement of this convergence result by providing a...

In this paper,  fuzzy convergence theory in the framework of $L$-convex spaces is introduced. Firstly, the concept of $L$-convex remote-neighborhood spaces is introduced and it is shown that the  resulting category is isomorphic to that of $L$-convex spaces. Secondly, by means of $L$-convex ideals, the notion of $L$-convergence spaces is introduced and it is proved that the  category of $L$-con...

in this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. furthermore,...